发表机构
United Arab Emirates University（阿联酋大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究具有两个中性不动点的平衡双间歇映射，证明在块支配采样下稀疏经验测度不收敛但具强自然测度性质，并验证对称二次映射的二进经验测度收敛到端点混合。
AI 中文摘要
我们考虑具有两个中性不动点的平衡双间歇全分支区间映射。尽管普通经验测度对Lebesgue-a.e.初始点不收敛，我们证明这种非统计行为在广泛的一类确定性观测方案下持续存在：如果采样序列是块支配的，则对Lebesgue-a.e.轨道，两个端点Dirac质量都是稀疏经验测度的聚点。该类包括正下密度采样序列、算术序列、多项式序列以及有限指标正则变化序列。我们还证明了平衡类中的每个映射都具有强自然测度性质，包括在边界指数$\beta=1$处：每个绝对连续初始概率的推送前向收敛到独特的端点混合$\nu_*$。这一一次性极限的抽象推论包括沿每个确定性采样序列的退火收敛、存在任意稀疏序列具有几乎必然收敛，以及基于幂节省方差界的几乎必然收敛准则。对于对称二次映射$g_1\in\mathfrak F_*$，我们在可数均匀稠密端点平坦族上验证了二进协方差条件：双重协方差和为$O_\phi(n\log(n+2))$。因此，其二进经验测度对Lebesgue-a.e.初始点收敛到$\tfrac12(\delta_{-1}+\delta_1)$。
英文摘要
We consider balanced doubly intermittent full-branch interval maps with two neutral fixed points. Although the ordinary empirical measures fail to converge for Lebesgue-almost every initial point, we show that this non-statistical behavior persists under a broad class of deterministic observation schemes: if the sampling sequence is block-dominating, then both endpoint Dirac masses are accumulation points of the sparse empirical measures for Lebesgue-almost every orbit. This class includes sampling sequences of positive lower density, arithmetic and polynomial sequences, and finite-index regularly varying sequences. We also prove that every map in the balanced class has the strong natural-measure property, including at the boundary exponent $β=1$: the pushforwards of every absolutely continuous initial probability converge to a distinguished endpoint mixture $ν_*$. Abstract consequences of this one-time limit include annealed convergence along every deterministic sampling sequence, the existence of arbitrarily sparse sequences with almost-sure convergence, and an almost-sure convergence criterion based on a power-saving variance bound. For the symmetric quadratic map $g_1\in\mathfrak F_*$, we verify the dyadic covariance condition on a countable uniformly dense endpoint-flat family: the double covariance sum is $O_ϕ(n\log(n+2))$. Consequently its dyadic empirical measures converge for Lebesgue-almost every initial point to $\tfrac12(δ_{-1}+δ_1)$.
Comments38 pages
